A prime-power cubic binomial supercongruence for primes congruent to 3 modulo 4

Let $p>3$ be a prime with $p\equiv3\pmod4$, and let $r$ be a positive integer. The supercongruence.

$$\sum_{k=0}^{p^r-1}\frac1{64^k}\sum_{j=0}^{k}{2j\choose j}^3{2k-2j\choose k-j}^3\equiv0\pmod{p^4}.$$

This is described as a generalization of an earlier case distinction; the stated modulus is independent of $r$, and the source gives no proof.